If Youtube videos won't display properly it's usually because of a problem with Adobe Flash Player (which is what your browser uses to display the videos).
Go to 'Programs & Features' (or whatever your version of Windows calls it) and uninstall 'Adobe Flash Player Plug-in'.
RESTART YOUR COMPUTER! (I've put it in capitals because it's IMPORTANT!).
Go here:
https://get.adobe.com/flashplayer/
DESELECT THE OPTION TO INSTALL McAFEE SECURITY SCAN PLUS. (That's IMPORTANT too!).
Click 'Install Now' and follow the instructions.
RESTART YOUR COMPUTER AGAIN. (Possibly not required but probably best to do so anyway).
Cross your fingers and try to view a Youtube video.